Categorical-algebraic aspects of Heyting semilattices

TL;DR

This paper explores the categorical-algebraic properties of Heyting semilattices, clarifying misconceptions, and establishing their structural features such as normality, commutativity, and the existence of centralisers, with implications for their algebraic coherence.

Contribution

It provides a detailed categorical-algebraic analysis of Heyting semilattices, correcting misconceptions and establishing new properties like normality of Higgins commutators and the existence of centralisers.

Findings

Heyting semilattices are not algebraically coherent.

Higgins commutators of normal subobjects are normal.

Centralisers of normal monomorphisms exist and are normal.

Abstract

This article gives an overview of some key categorical-algebraic properties of the variety of Heyting semilattices, with the aim of correcting a misconception in the literature. We confirm that the category of Heyting semilattices is not algebraically coherent, even though it satisfies a strong version of the so-called Smith is Huq condition (on the equivalence of two types of commutators). We also prove that Higgins commutators of normal subobjects are normal, as a consequence of the fact that Heyting semilattices form an arithmetical category. We provide an elementary characterisation of when a pair of subobjects commutes, and use this in the construction of two counterexamples. We further show that centralisers exist, centralisers of normal monomorphisms are normal monomorphisms, and normal monomorphisms are closed under composition. We study the latter condition in detail. On the other hand, we show that the category of Heyting semilattices does not satisfy normality of unions. Hence, it is not action accessible and so it does not admit all normalisers. In particular, this means that the known implication between action accessibility and the condition requiring the existence of centralisers of normal monomorphisms which are themselves normal, is strict.